EEE 498/598 Overview of Electrical Engineering

1. EEE 498/598Overview of Electrical Engineering Lecture 2: Introduction to Electromagnetic Fields; Maxwell’s Equations; Electromagnetic Fields in Materials; Phasor Concepts; Electrostatics: Coulomb’s Law, Electric Field, Discrete and Continuous Charge Distributions; Electrostatic Potential 1

2. Lecture 2 Objectives • To provide an overview of classical electromagnetics, Maxwell’s equations, electromagnetic fields in materials, and phasor concepts. • To begin our study of electrostatics with Coulomb’s law; definition of electric field; computation of electric field from discrete and continuous charge distributions; and scalar electric potential. 2

3. Introduction to Electromagnetic Fields • Electromagnetics is the study of the effect of charges at rest and charges in motion. • Some special cases of electromagnetics: • Electrostatics: charges at rest • Magnetostatics: charges in steady motion (DC) • Electromagnetic waves: waves excited by charges in time-varying motion 3

4. Circuit Theory Introduction to Electromagnetic Fields Maxwell’s equations Fundamental laws of classical electromagnetics Geometric Optics Electro-statics Magneto-statics Electro-magnetic waves Special cases Statics: Transmission Line Theory Input from other disciplines Kirchoff’s Laws 4

5. Introduction to Electromagnetic Fields • transmitter and receiver • are connected by a “field.” 5

6. 2 3 1 4 Introduction to Electromagnetic Fields High-speed, high-density digital circuits: • consider an interconnect between points “1” and “2” 6

7. Introduction to Electromagnetic Fields • Propagation delay • Electromagnetic coupling • Substrate modes 7

8. Introduction to Electromagnetic Fields • When an event in one place has an effect on something at a different location, we talk about the events as being connected by a “field”. • A fieldis a spatial distribution of a quantity; in general, it can be either scalar or vector in nature. 8

9. Introduction to Electromagnetic Fields • Electric and magnetic fields: • Are vector fields with three spatial components. • Vary as a function of position in 3D space as well as time. • Are governed by partial differential equations derived from Maxwell’s equations. 9

10. Introduction to Electromagnetic Fields • A scalaris a quantity having only an amplitude (and possibly phase). • A vectoris a quantity having direction in addition to amplitude (and possibly phase). Examples: voltage, current, charge, energy, temperature Examples: velocity, acceleration, force 10

11. Introduction to Electromagnetic Fields • Fundamental vector field quantities in electromagnetics: • Electric field intensity • Electric flux density (electric displacement) • Magnetic field intensity • Magnetic flux density units = volts per meter (V/m = kg m/A/s3) units = coulombs per square meter (C/m2 = A s /m2) units = amps per meter (A/m) units = teslas = webers per square meter (T = Wb/ m2 = kg/A/s3) 11

12. Introduction to Electromagnetic Fields • Universal constants in electromagnetics: • Velocity of an electromagnetic wave (e.g., light) in free space (perfect vacuum) • Permeability of free space • Permittivity of free space: • Intrinsic impedance of free space: 12

13. Introduction to Electromagnetic Fields • Relationships involving the universal constants: In free space: 13

14. sources Ji, Ki fields E, H Introduction to Electromagnetic Fields • Obtained • by assumption • from solution to IE Solution to Maxwell’s equations Observable quantities 14

15. Maxwell’s Equations • Maxwell’s equations in integral form are the fundamental postulates of classical electromagnetics - all classical electromagnetic phenomena are explained by these equations. • Electromagnetic phenomena include electrostatics, magnetostatics, electromagnetostatics and electromagnetic wave propagation. • The differential equations and boundary conditions that we use to formulate and solve EM problems are all derived from Maxwell’s equations in integral form. 15

16. Maxwell’s Equations • Various equivalence principles consistent with Maxwell’s equations allow us to replace more complicated electric current and charge distributions with equivalent magnetic sources. • These equivalent magnetic sources can be treated by a generalization of Maxwell’s equations. 16

17. Maxwell’s Equations in Integral Form (Generalized to Include Equivalent Magnetic Sources) Adding the fictitious magnetic source terms is equivalent to living in a universe where magnetic monopoles (charges) exist. 17

18. Continuity Equation in Integral Form (Generalized to Include Equivalent Magnetic Sources) • The continuity equations are implicit in Maxwell’s equations. 18

19. S C dS S V dS Contour, Surface and Volume Conventions • open surface S bounded by • closed contour C • dS in direction given by • RH rule • volume V bounded by • closed surface S • dS in direction outward • from V 19

20. Electric Current and Charge Densities • Jc = (electric) conduction current density (A/m2) • Ji = (electric) impressed current density (A/m2) • qev = (electric) charge density (C/m3) 20

21. Magnetic Current and Charge Densities • Kc = magnetic conduction current density (V/m2) • Ki = magnetic impressed current density (V/m2) • qmv = magnetic charge density (Wb/m3) 21

22. Maxwell’s Equations - Sources and Responses • Sources of EM field: • Ki, Ji, qev, qmv • Responses to EM field: • E, H, D, B, Jc, Kc 22

23. Maxwell’s Equations in Differential Form (Generalized to Include Equivalent Magnetic Sources) 23

24. Continuity Equation in Differential Form (Generalized to Include Equivalent Magnetic Sources) • The continuity equations are implicit in Maxwell’s equations. 24

25. Electromagnetic Boundary Conditions Region 1 Region 2 25

26. Electromagnetic Boundary Conditions 26

27. Surface Current and Charge Densities • Can be either sources of or responses to EM field. • Units: • Ks - V/m • Js - A/m • qes - C/m2 • qms - W/m2 27

28. Electromagnetic Fields in Materials • In time-varying electromagnetics, we consider E and H to be the “primary” responses, and attempt to write the “secondary” responses D, B, Jc, and Kcin terms of E and H. • The relationships between the “primary” and “secondary” responses depends on the medium in which the field exists. • The relationships between the “primary” and “secondary” responses are called constitutive relationships. 28

29. Electromagnetic Fields in Materials • Most general constitutive relationships: 29

30. Electromagnetic Fields in Materials • In free space, we have: 30

31. Electromagnetic Fields in Materials • In a simple medium, we have: • linear(independent of field strength) • isotropic(independent of position within the medium) • homogeneous(independent of direction) • time-invariant(independent of time) • non-dispersive (independent of frequency) 31

32. Electromagnetic Fields in Materials • e = permittivity = ere0 (F/m) • m = permeability = mrm0(H/m) • s = electric conductivity = ere0(S/m) • sm = magnetic conductivity = ere0(W/m) 32

33. Phasor Representation of a Time-Harmonic Field • A phasor is a complex number representing the amplitude and phase of a sinusoid of known frequency. phasor frequency domain time domain 33

34. Phasor Representation of a Time-Harmonic Field • Phasors are an extremely important concept in the study of classical electromagnetics, circuit theory, and communications systems. • Maxwell’s equations in simple media, circuits comprising linear devices, and many components of communications systems can all be represented as linear time-invariant (LTI) systems. (Formal definition of these later in the course …) • The eigenfunctions of any LTI system are the complex exponentials of the form: 34

35. If the input to an LTI system is a sinusoid of frequency w, then the output is also a sinusoid of frequency w (with different amplitude and phase). Phasor Representation of a Time-Harmonic Field A complex constant (for fixed w); as a function of w gives the frequency response of the LTI system. 35

36. Phasor Representation of a Time-Harmonic Field • The amplitude and phase of a sinusoidal function can also depend on position, and the sinusoid can also be a vector function: 36

37. Phasor Representation of a Time-Harmonic Field • Given the phasor (frequency-domain) representation of a time-harmonic vector field, the time-domain representation of the vector field is obtained using the recipe: 37

38. Phasor Representation of a Time-Harmonic Field • Phasors can be used provided all of the media in the problem are linear no frequency conversion. • When phasors are used, integro-differential operators in time become algebraic operations in frequency, e.g.: 38

39. Time-Harmonic Maxwell’s Equations • If the sources are time-harmonic (sinusoidal), and all media are linear, then the electromagnetic fields are sinusoids of the same frequency as the sources. • In this case, we can simplify matters by using Maxwell’s equations in the frequency-domain. • Maxwell’s equations in the frequency-domain are relationships between the phasor representations of the fields. 39

40. Maxwell’s Equations in Differential Form for Time-Harmonic Fields 40

41. Maxwell’s Equations in Differential Form for Time-Harmonic Fields in Simple Medium 41

42. Circuit Theory Electrostatics as a Special Case of Electromagnetics Maxwell’s equations Fundamental laws of classical electromagnetics Geometric Optics Electro-statics Magneto-statics Electro-magnetic waves Special cases Statics: Transmission Line Theory Input from other disciplines Kirchoff’s Laws 42

43. Electrostatics • Electrostaticsis the branch of electromagnetics dealing with the effects of electric charges at rest. • The fundamental law of electrostatics is Coulomb’s law. 43

44. Electric Charge • Electrical phenomena caused by friction are part of our everyday lives, and can be understood in terms of electrical charge. • The effects of electrical charge can be observed in the attraction/repulsion of various objects when “charged.” • Charge comes in two varieties called “positive” and “negative.” 44

45. Electric Charge • Objects carrying a net positive charge attract those carrying a net negative charge and repel those carrying a net positive charge. • Objects carrying a net negative charge attract those carrying a net positive charge and repel those carrying a net negative charge. • On an atomic scale, electrons are negatively charged and nuclei are positively charged. 45

46. Electric Charge • Electric charge is inherently quantized such that the charge on any object is an integer multiple of the smallest unit of charge which is the magnitude of the electron charge e= 1.602  10-19C. • On the macroscopic level, we can assume that charge is “continuous.” 46

47. Coulomb’s Law • Coulomb’s law is the “law of action” between charged bodies. • Coulomb’s law gives the electric force between two point chargesin an otherwise empty universe. • A point charge is a charge that occupies a region of space which is negligibly small compared to the distance between the point charge and any other object. 47

48. Coulomb’s Law Q1 Q2 Unit vector in direction of R12 Force due to Q1 acting on Q2 48

49. Coulomb’s Law • The force on Q1due to Q2 is equal in magnitude but opposite in direction to the force on Q2 due to Q1. 49

50. Qt Q Electric Field • Consider a point charge Q placed at the origin of a coordinate system in an otherwise empty universe. • A test charge Qtbrought near Qexperiences a force: 50